Lost in the middle class

Washington Post asks people what it means to be middle class in the U.S. (link; paywall)

The following graphic illustrates one type of definition, purely based on income ranges.

Wpost_middleclass

For me, this chart is more taxing to read than it appears.

It can be read column by column. Each column represents a hypotheticial annual income for a family of four. People are asked whether they consider that family lower/working class, middle class or upper class. Be careful as the increments from column to column are not uniform.

Now, what's the question again? We're primarily interested in what incomes constitute middle class.

So, we should be looking at the deep green blocks that hang in the middle of each column. It's not easy to read the proportion of middle blocks in a stacked column chart.

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I tried separating out the three perceived income classes, using a small-multiples design.

Junkcharts_redo_wpost_middleclass

One can more directly see what income ranges are most popularly perceived as being in each income class.

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The article also goes into alternative definitions of middle class, using more qualitative metrics, such as "able to pay all bills on time without worry". That's a whole other post.

 


Neither the forest nor the trees

On the NYT's twitter feed, they featured an article titled "These Seven Tech Stocks are Driving the Market". The first sentence of the article reads: "The S&P 500 is at an all-time high, and investors have just a handful of stocks to thank for it."

Without having seen any data, I'd surmise from that line that (a) the S&P 500 index has gone up recently, and (b) most if not all of the gain in the index can be attributed to gains in the tech stocks mentioned in the headline. (For purists, a handful is five, not seven.)

The chart accompanying the tweet is a treemap:

Nyt_magnificentseven

The treemap is possibly the most overhyped chart type of the modern era. Its use here is tangential to the story of surging market value. That's because the treemap presents a snapshot of the composition of the index, but contains nothing about the trend (change over time) of the average index value or of its components.

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Even in representing composition, the treemap is inferior to, gasp, a pie chart. Of course, we can only use a pie chart for small numbers of components. The following illustration takes the data from the NYT chart on the Magnificent Seven tech stocks, and compares a treemap versus a pie chart side by side:

Junkcharts_redo_nyt_magnificent7

The reason why the treemap is worse is that both the width and the height of the boxes are changing while only the radius (or angle) of the pie slices is varying. (Not saying use a pie chart, just saying the treemap is worse.)

There is a reason why the designer appended data labels to each of the seven boxes. The effect of not having those labels is readily felt when our eyes reach the next set of stocks – which carry company names but not their market values. What is the market value of Berkshire Hathaway?

Even more so, what proportion of the total is the market value of Berkshire Hathaway? Indeed, if the designer did not write down 29%, it would take a bit of work to figure out the aggregate value of yellow boxes relative to the entire box!

This design sucessfully draws our attention to the structural importance of various components of the whole. There are three layers - the yellow boxes (Magnificent Seven), the gray boxes with company names, and the other gray boxes. I also like how they positioned the text on the right column.

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Going inside the NYT article itself, we find two line charts that convey the story as told.

Here's the first one:

Nyt_magnificent7_linechart1

They are comparing the most recent stock prices with those from October 12 2022, which is identified as the previous "low". (I'm actually confused by how the most recent "low" is defined, but that's a different subject.)

This chart carries a lot of good information, even though it does not plot "all the data", as in each of the 500 S&P components individually. Over the period under analysis, the average index value has gone up about 35% while the Magnificent Seven's value have skyrocketed by 65% in aggregate. The latter accounted for 30% of the total value at the most recent time point.

If we set the S&P 500 index value in 2024 as 100, then the M7 value in 2024 is 30. After unwinding the 65% growth, the M7 value in October 2022 was 18; the S&P 500 in October 2022 was 74. Thus, the weight of M7 was 24% (18/74) in October 2022, compared to 30% now. Consequently, the weight of the other 473 stocks declined from 76% to 70%.

This isn't even the full story because most of the action within the M7 is in Nvidia, the stock most tightly associated with the current AI hype, as shown in the other line chart.

Nyt_magnificent7_linechart2

Nvidia's value jumped by 430% in that time window. From the treemap, the total current value of M7 is $12.3 b while Nvidia's value is $1.4 b, thus Nvidia is 11.4% of M7 currently. Since M7 is 29% of the total S&P 500, Nvidia is 11.4%*29% = 3% of the S&P. Thus, in 2024, against 100 for the S&P, Nvidia's share is 3. After unwinding the 430% growth, Nvidia's share in October 2022 was 0.6, about 0.8% of 74. Its weight tripled during this period of time.


The cult of raw unadjusted data

Long-time reader Aleks came across the following chart on Facebook:

Unadjusted temp data fgfU4-ia fb post from aleks

The author attached a message: "Let's look at raw, unadjusted temperature data from remote US thermometers. What story do they tell?"

I suppose this post came from a climate change skeptic, and the story we're expected to take away from the chart is that there is nothing to see here.

***

What are we looking at, really?

"Nothing to see" probably refers to the patch of blue squares that cover the entire plot area, as time runs left to right from the 1910s to the present.

But we can't really see what's going on in the middle of the patch. So, "nothing to see" is effectively only about the top-to-bottom range of roughly 29.8 to 82.0. What does that range signify?

The blue patch is subdivided into vertical lines consisting of blue squares. Each line is a year's worth of temperature measurements. Each square is the average temperature on a specific day. The vertical range is the difference between the maximum and minimum daily temperatures in a given year. These are extreme values that say almost nothing about the temperatures in the other ~363 days of the year.

We know quite a bit more about the density of squares along each vertical line. They are broken up roughly by seasons. Those values near the top came from summers while the values near the bottom came from winters. The density is the highest near the middle, where the overplotting is so severe that we can barely see anything.

Within each vertical line, the data are not ordered chronologically. This is a very key observation. From left to right, the data are ordered from earliest to latest but not from top to bottom! Therefore, it is impossible for the human eye to trace the entire trajectory of the daily temperature readings from this chart. At best, you can trace the yearly average temperature – but only extremely roughly by eyeballing where the annual averages are inside the blue patch.

Indeed, there is "nothing to see" on this chart because its design has pulverized the data.

***

_numbersense_bookcoverIn Numbersense (link), I wrote "not adjusting the raw data is to knowingly publish bad information. It is analogous to a restaurant's chef knowingly sending out spoilt fish."

It's a fallacy to think that "raw unadjusted" data are the best kind of data. It's actually the opposite. Adjustments are designed to correct biases or other problems in the data. Of course, adjustments can be subverted to introduce biases in the data as well. It is subversive to presume that all adjustments are of the subversive kind.

What kinds of adjustments are of interest in this temperature dataset?

Foremost is the seasonal adjustment. See my old post here. If we want to learn whether temperatures have risen over these decades, we can't do so without separating out the seasons.

The whole dataset can be simplified by drawing the smoothed annual average temperature grouped by season of the year, and when that is done, the trend of rising temperatures is obvious.

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The following chart by the EPA roughly implements the above:

Epa-seasonal-temperature_2022

The original can be found here. They made one adjustment which isn't the one I expected.

Note the vertical scale is titled "temperature anomaly". So, they are not plotting the actual recorded average temperatures, but the "anomalies", i.e. the difference between the recorded temperatures and some kind of "expected" temperature. This is a type of data adjustment as well. The purpose is to focus attention on the relative rather than absolute values. Think of this formula: recorded value = expected value + anomaly. The chart shows how many degrees above or below expectation, rather than how many degrees.

For a chart like this, there should be a required footnote that defines what "anomaly" is. Specifically, the reader should know about the model behind the "expectation". Typically, it's a kind of long-term average value.

For me, this adjustment is not necessary. Without the adjustment, the four panels can be combined into one panel with four lines. That's because the data nicely fit into four levels based on seasons.

The further adjustment I'd have liked to see is "smoothing". Each line above has a "smooth" trend, as well as some variability around this trend. The latter is not a big part of the story.

***

It's weird to push back on climate change advocacy by attacking data adjustments. The more productive direction, in my view, is to ask whether the observed trend is caused by human activities or part of some long-term up-and-down cycle. That is a very challenging question to answer.


To a new year of pleasant surprises

Happy new year!

This year promises to be the year of AI. Already last year, we pretty much couldn't lift an eyebrow without someone making an AI claim. This year will be even noisier. Visual Capitalist acknowledged this by making the noisiest map of 2023:

Visualcapitalist_01_Generative_AI_World_map sm

I kept thinking they have a geography teacher on the team, who really, really wants to give us a lesson of where each country is on the world map.

All our attention is drawn to the guiding lines and the random scatter of numbers. We have to squint to find the country names. All this noise drowns out the attempt to make sense of the data, namely, the inset of the top 10 countries in the lower left corner, and the classification of countries into five colored groups.

A small dose of editing helps. Remove most data labels except for the countries for which they have a story. Provide a data table below for those who want details.

***

In the Methodology section, the data analysts (possibly from a third party called ElectronicsHub) indicated that they used Google search volume of "over 90 of the most popular generative AI tools", calculating the "overall volume across all tools per 100k population". Then came a baffling line: "all search volumes were scaled up according to the search engine market share in each country, using figures from statscounter.com." (Note: in the following, I'm calling the data "AI-related search" for simplicity even though their measurement is restricted to the terms described above.)

It took me a while to comprehend what they could have meant by that line. I believe this is what that sentence means: Google is not the only search engine out there so by only researching Google search volume, they undercount the true search volume. How did they deal with the missing data problem? They "scaled up" so if Google is 80% of the search volume in a country, then they divide the Google volume by 80% to "scale up" to 100%.

Whenever we use heuristics like this, we should investigate its foundations. What is the implicit assumption behind this scaling-up procedure? It is that all search engines are effectively the same. The users of non-Google search engines behave exactly as the Google search engine users. If the analysts somehow could get their hands on the data of other search engines, they would discover that the proportion of search volume that is AI-related is effectively the same as seen on Google.

This is one of those convenient, and obviously wrong assumptions – if true, the market would have no need for more than one search engine. Each search engine's audience is just a random sample from the population of all users.

Let's make up some numbers. Let's say Google has 80% share of search volume in Country A, and AI-related search 10% of the overall Google search volume. The remaining search engines have 20% share. Scaling up here means taking the 8% of Google AI-related search volume, divide by 80%, which yields 10%. Since Google owns 8% of the 10%, the other search engines see 2% of overall search volume attributed to AI searches in Country A. Thus, the proportion of AI-related searches on those other search engines is 2%/20% = 10%.

Now, in certain countries, Google is not quite as dominant. Let's say Google only has 20% share of Country B's search volume. AI-related search on Google is 2%, which is 10% of its total. Using the same scaling-up procedure, the analysts have effectively assumed that the proportion of AI-related search volume in the dominant search engines in Country B to be also 10%.

I'm using the above calculations to illustrate a shortcoming of this heuristic. Using this procedure inflates the search volume in countries in which Google is less dominant because the inflation factor is the reciprocal of Google's market share. The less dominant Google is, the larger the inflation factor.

What's also true? The less dominant Google is, the smaller proportion of the total data the analysts are able to see, the lower the quality of the available information. So the heuristic is the most influential where it has the greatest uncertainty.

***

Hope your new year is full of uncertainty, and your heuristics shall lead you to pleasant surprises.

If you like the blog's content, please spread the word. I'm looking forward to sharing more content as the world of data continues to evolve at an amazing pace.

Disclosure: This blog post is not written by AI.


The efficiency of visual communications

Visual Capitalist has this wonderful chart showing the gaps between the stock market returns expected by "investors" compared to "professionals".

Visualcapitalism_Global-Investor-Gap_11172023It's a model of clarity. The chart form is a dot plot.

The blue dots represent what investors (individuals?) expect to earn from investing in the stock market in the long run. The orange dots represent the professional viewpoint. Each row shows survey results in a different country.

At first glance, U.S. investors are vastly more optimistic than professionals. There is excess enthusiasm in most other countries as well.

The exceptions are Chile, Mexico and Singapore in which the two groups are almost perfectly aligned. The high degree of concordance in these countries makes me wonder if their investors are demographically similar to professionals.

***

Those are the first insights one can take from the dot plot, with almost no effort.

But there's more.

The global average is shown in the middle of the chart, allowing readers to compare each country against it. (It's not clear what average this represents though - maybe the average return expected by investors?)

There's more.

Junkcharts_redo_visualcapitalist_expectedreturnsgapsThe chart shows what's professional about professionals. Not only do professionals hold a much more pessimistic view of stock returns in general, they also exhibit a much lower variance in expectations.

This reflects that professionals adhere to an orthodoxy - they went to the same schools, were taught from the same textbooks, took the same professional exams, and live in their own echo chambers.

Chile, Mexico and Singapore, however, stick out. For a change, the professionals share the enthusiasm of investors.

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This chart shows the power of data visualization. So much information can be conveyed in a small space, if one designs the visual well.


Stranger things found on scatter plots

Washington Post published a nice scatter plot which deconstructs scores from the recent World Championships in Gymnastics. (link)

Wpost_simonebiles

The chart presents the main message clearly - the winner Simone Biles scored the highest on both components of the score (difficulty and execution), by quite some margin.

What else can we learn from this chart?

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Every athlete who qualified for the final scored at or above average on both components.

Scoring below average on either component is a death knell: no athlete scored enough on the other component to compensate. (The top left and bottom right quadrants would have had some yellow dots otherwise.)

Several athletes in the top right quadrant presumably scored enough to qualify but didn't. The footnote likely explains it: each country can send at most two athletes to the final. It may be useful to mark out these "unlucky" athletes using a third color.

Curiously, it's not easy to figure out who these unlucky athletes were from this chart alone. We need two pieces of data: the minimum qualifying score, and the total score for each athlete. The scatter plot isn't the best chart form to show totals, but qualification to the final is based on the sum of the difficulty and execution scores. (Note also, neither axis starts at zero, compounding the challenge.)

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This scatter plot is most memorable for shattering one of my expectations about risk and reward in sports.

I expect risk-seeking athletes to suffer from higher variance in performance. The tennis player who goes for big serves tend to also commit more double faults. The sluggers who hit home runs tend to strike out more often. Similarly, I expect gymnasts who attempt more difficult skills to receive lower execution scores.

Indeed, the headline writer seemed to agree, suggesting that Biles is special because she's both high in difficulty and strong in execution.

The scatter plot, however, sends the opposite message - this should not surprise. The entire field shows a curiously strong positive correlation between difficulty and execution scores. The more difficult is the routine, the higher the excution score!

It's hard to explain such a pattern. My guesses are:

a) judges reward difficult routines, and subconsciously confound execution and difficulty scores. They use separate judges for excecution and difficulty. Paradoxically, this arrangement may have caused separation anxiety - the judges for execution might just feel the urge to reward high difficulty.

b) those athletes who are skilled enough to attempt more difficult routines are also those who are more consistent in execution. This is a type of self-selection bias frequently found in observational data.

Regardless of the reasons for the strong correlation, the chart shows that these two components of the total score are not independent, i.e. the metrics have significant overlap in what they measure. Thus, one cannot really talk about a difficult routine without also noting that it's a well-executed routine, and vice versa. In an ideal scoring design, we'd like to have independent components.


The choice to encode data using colors

NBC News published the following heatmap that shows inflation by product category in the last year or so:

Nbcnews_inflationtracker

The general story might be that inflation was rampant in airfare and electricity prices about a year ago but these prices have moderated recently, especially in airfare. Gas prices appear to have inflated far less than overall inflation during these months.

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Now, if you're someone who cares about the magnitude of differences, not just the direction, then revisit the above statements, and you'll feel a sense of inadequacy.

When we choose to encode data in colors, we're giving up on showing magnitudes or precision. The color scale shown up top sends the message that the continuous nature of the number line is being displayed but it really isn't.

The largest value of the chart is found on the left side of the airfare row:

Nbcnews_inflationtracker_highest

The value is about 36% which strangely enough is far larger than the maximum value shown in the legend above. Even if those values align, it is still impossible to guess what values the different colors and shades in the cells map to from the legend.

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The following small-multiples chart shows the underlying values more precisely:

Redo_junkcharts_nbcnewsinflation

I have transformed the data differently. In these line charts, the data are indexed to the first month (100) so each chart shows the cumulative change in prices from that month to the current month, for each category, compared to the overall.

The two most interesting categories are airfare and gas. Airfare has recently decreased quite drastically relative to September 2022, and thus the line is far below the overall inflation trend. Gas prices moved in reverse: they dropped in the last quarter of 2022 but have steadily risen over 2023, and in the most recent month, is tracking overall inflation.

 

 


What is the question is the question

I picked up a Fortune magazine while traveling, and saw this bag of bubbles chart.

Fortune_global500 copy

This chart is visually appealing, that must be said. Each circle represents the reported revenues of a corporation that belongs to the “Global 500 Companies” list. It is labeled by the location of the company’s headquarters. The largest bubble shows Beijing, the capital of China, indicating that companies based in Beijing count $6 trillion dollars of revenues amongst them. The color of the bubbles show large geographical units; the red bubbles are cities in Greater China.

I appreciate a couple of the design decisions. The chart title and legend are placed on the top, making it easy to find one’s bearing – effective while non-intrusive. The labeling signals a layering: the first and biggest group have icons; the second biggest group has both name and value inside the bubbles; the third group has values inside the bubbles but names outside; the smallest group contains no labels.

Note the judgement call the designer made. For cities that readers might not be familiar with, a country name (typically abbreviated) is added. This is a tough call since mileage varies.

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As I discussed before (link), the bag of bubbles does not elevate comprehension. Just try answering any of the following questions, which any of us may have, using just the bag of bubbles:

  • What proportion of the total revenues are found in Beijing?
  • What proportion of the total revenues are found in Greater China?
  • What are the top 5 cities in Greater China?
  • What are the ranks of the six regions?

If we apply the self-sufficiency test and remove all the value labels, it’s even harder to figure out what’s what.

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_trifectacheckup_image

Moving to the D corner of the Trifecta Checkup, we aren’t sure how to interpret this dataset. It’s unclear if these companies derive most of their revenues locally, or internationally. A company headquartered in Washington D.C. may earn most of its revenues in other places. Even if Beijing-based companies serve mostly Chinese customers, only a minority of revenues would be directly drawn from Beijing. Some U.S. corporations may choose its headquarters based on tax considerations. It’s a bit misleading to assign all revenues to one city.

As we explore this further, it becomes clear that the designer must establish a target – a strong idea of what question s/he wants to address. The Fortune piece comes with a paragraph. It appears that an important story is the spatial dispersion of corporate revenues in different countries. They point out that U.S. corporate HQs are more distributed geographically than Chinese corporate HQs, which tend to be found in the key cities.

There is a disconnect between the Question and the Data used to create the visualization. There is also a disconnect between the Question and the Visual display.


Partition of Europe

A long-time reader sent me the following map via twitter:

Europeelects_map

This map tells how the major political groups divide up the European Parliament. I’ll spare you the counting. There are 27 countries, and nine political groups (including the "unaffiliated").

The key chart type is a box of dots. Each country gets its own box. Each box has its own width. What determines the width? If you ask me, it’s the relative span of the countries on the map. For example, the narrow countries like Ireland and Portugal have three dots across while the wider countries like Spain, Germany and Italy have 7, 10 and 8 dots across respectively.

Each dot represents one seat in the Parliament. Each dot has one of 9 possible colors. Each color shows a political lean e.g. the green dots represent Green parties while the maroon dots display “Left” parties.

The end result is a counting game. If we are interested in counts of seats, we have to literally count each dot. If we are interested in proportion of seats, take your poison: either eyeball it or count each color and count the total.

Who does the underlying map serve? Only readers who know the map of Europe. If you don’t know where Hungary or Latvia is, good luck. The physical constraints of the map work against the small-multiples set up of the data. In a small multiples, you want each chart to be identical, except for the country-specific data. The small-multiples structure requires a panel of equal-sized cells. The map does not offer this feature, as many small countries are cramped into Eastern Europe. Also, Europe has a few tiny states e.g. Luxembourg (population 660K)  and Malta (population 520K). To overcome the map, the designer produces boxes of different sizes, substantially loading up the cognitive burden on readers.

The map also dictates where the boxes are situated. The centroids of each country form the scaffolding, with adjustments required when the charts overlap. This restriction ensures a disorderly appearance. By contrast, the regular panel layout of a small multiples facilitates comparisons.

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Here is something I sketched using a tile map.

Eu parties print sm

First, I have to create a tile map of European countries. Some parts, e.g. western part, are straightforward. The eastern side becomes very congested.

The tile map encodes location in an imprecise sense. Think about the scaffolding of centroids of countries referred to prior. The tile map imposes an order to the madness - we're shifting these centroids so that they line up in a tidier pattern. What we gain in comparability we concede in location precision.

For the EU tile map, I decided to show the Baltic countries in a row rather than a column; the latter would have been more faithful to the true geography. Malta is shown next to Italy even though it could have been placed below. Similarly, Cyprus in relation to Greece. I also included several key countries that are not part of the EU for context.

Instead of raw seat counts, I'm showing the proportion of seats within each country claimed by each political group. I think this metric is more useful to readers.

The legend is itself a chart that shows the aggregate statistics for all 27 countries.


Tile maps on a trip

My friend Ray sent me to a recent blog about tile maps. Typical tile maps use squares or hexagons, although in theory many other shapes will do. Unsurprisingly, the field follows the latest development of math researchers who study the space packing problem. The space packing problem concerns how to pack a space with objects. The study of tesselations is to pack space with one or a few shapes.

It was an open question until recently whether there exists an "aperiodic monotile," that is to say, a single shape that can cover space in a non-repeating manner. We all know that we can use squares to cover a space, which creates the familiar grid of squares, but in that case, a pattern repeats itself all over the space.

Now, some researchers have found an elusive aperiodic monotile, which they dubbed the Einstein monotile. Below is a tesselation using these tiles:

Einsteintiles

Within this design, one cannot find a set of contiguous tiles that repeats itself.

The blogger then made a tile map using this new tesselation. Here's one:

Gravitywitheinsteintiles

It doesn't matter what this is illustrating. The blog author cites a coworker, who said: "I can think of no proper cartographic use for Penrose binning, but it’s fun to look at, and so that’s good enough for me." Penrose tiles is another mathematical invention that can be used in a tesselation. The story is still the same: there is no benefit from using these strange-looking shapes. Other than the curiosity factor.

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Let's review the pros and cons of using tile maps.

Compare a typical choropleth map of the United States (by state) and a tile map by state. The former has the well-known problem that states with the largest areas usually have the lowest population densities, and thus, if we plot demographic data on such maps, the states that catch the most attention are the ones that don't weigh as much - by contrast, the densely populated states in New England barely show up.

The tile map removes this area bias, thus resolving this problem. Every state is represented by equal area.

While the tesselated design is frequently better, it's not always. In many data visualization, we do intend to convey the message that not all states are equal!

The grid arrangement of the state tiles also makes it easier to find regional patterns. A regional pattern is defined here as a set of neighboring states that share similar data (encoded in the color of the tiles). Note that the area of each state is of zero interest here, and thus, the accurate descriptions of relative areas found on the usual map is a distractor.

However, on the tile map, these regional patterns are conceptual. One must not read anything into the shape of the aggregated region, or its boundaries. Indeed, if we use strange-looking shapes like Einstein tiles, the boundaries are completely meaningless, and even misleading.

There also usually is some distortion of the spatial coordinates on a tile map because we'd like to pack the squares or hexagons into a lattice-like structure.

Lastly, the tile map is not scalable. We haven't seen a tile map of the U.S. by county or precinct but we have enjoyed many choropleth maps displaying county- or precinct-level data, e.g. the famous Purple Map of America. There is a reason for this.

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Here is an old post that contains links to various other posts I've written about tile maps.